
Yaml
aliases:
  - Eigenvectors
  - Eigenvalue
  - Eigenvector
  - Eigenvalue
  - Eigenspace
  - Eigenspaces

Eigenvectors and Eigenvalues

In a transformation, eigenvectors are the vectors that don't change direction, but get just scaled.

The factors by which they get scaled is called eigenvalue, and each eigenvactor has an associated eigenvalue.

There can also be no eigenvectors.

Example

Imagine we have two vectors in the basic plane:

We put the plane through the transformation:

Every other vector drifts to another direction, but not these guys.

These vectors just get scaled.

Info

The determinant of a matrix is the product of eigenvalues.

Info

In a 3D rotation, the axis of rotation is an eigenvector. It doesn't change direction.

Formal definition

The eigen vectors are such that when receiving a linear transformation, they stay on the same original direction, only scaled by their eigenvalue().

This process must be done for each eigenvalue separately.

We input in this equations for each eigenvalue.
We are left with a system of linear equations that we can solve using any method we want.

Now we solve the system:

Now we assign to